Hahn distance: Difference between revisions
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In {{w|Graph (mathematics)|graph theory}}, the {{w|Distance (graph theory)|distance}} between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path; if there is no path connection them, the distance is regarded as infinite. Given a set of [[just interval]]s, or more usually, of [[pitch class|classes of octave-equivalent intervals]], we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a [[consonance]]. Normally the [[unison]] is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales. | In {{w|Graph (mathematics)|graph theory}}, the {{w|Distance (graph theory)|distance}} between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path; if there is no path connection them, the distance is regarded as infinite. Given a set of [[just interval]]s, or more usually, of [[pitch class|classes of octave-equivalent intervals]], we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a [[consonance]]. Normally the [[unison]] is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales. | ||
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\begin{align} | \begin{align} | ||
& \lVert 3^a \cdot 5^b \cdot 7^c \rVert_\text {hahn} \\ | & \lVert 3^a \cdot 5^b \cdot 7^c \rVert_\text {hahn} \\ | ||
=& (\lvert a \rvert + \lvert b \rvert + \lvert c \rvert + \lvert a + b + c \rvert)/2 \\ | =& \left(\lvert a \rvert + \lvert b \rvert + \lvert c \rvert + \lvert a + b + c \rvert\right)/2 \\ | ||
=& \max(\lvert a \rvert, \lvert b \rvert, \lvert c \rvert, \lvert a + b \rvert, \lvert b + c \rvert, \lvert c + a \rvert, \lvert a + b + c \rvert) | =& \max\left(\lvert a \rvert, \lvert b \rvert, \lvert c \rvert, \lvert a + b \rvert, \lvert b + c \rvert, \lvert c + a \rvert, \lvert a + b + c \rvert\right) | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
We may take this formula and apply it to any triple of real numbers ‖(a, b, c)‖<sub>hahn</sub> = | We may take this formula and apply it to any triple of real numbers {{nowrap|‖(''a'', ''b'', ''c'')‖<sub>hahn</sub> {{=}} {{sfrac|{{!}}''a''{{!}} + {{!}}''b''{{!}} + {{!}}''c''{{!}} + {{!}}''a'' + ''b'' + ''c''{{!}}|2}}}}. | ||
If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice; it also defines a seminorm on 7-limit [[Monzos_and_Interval_Space|interval space]]. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by | If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice; it also defines a seminorm on 7-limit [[Monzos_and_Interval_Space|interval space]]. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by | ||
<math>\displaystyle \left\lVert (a, b, c) \right\rVert_\text {sym} = \sqrt{ | <math>\displaystyle \left\lVert (a, b, c) \right\rVert_\text {sym} = \sqrt{a^2 + b^2 + c^2 + ab + bc + ca}</math> | ||
and discussed in [[The Seven Limit Symmetrical Lattices]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice. | and discussed in [[The Seven Limit Symmetrical Lattices]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice. | ||
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\begin{align} | \begin{align} | ||
& \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert_\text{hahn} \\ | & \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert_\text{hahn} \\ | ||
=& (\lvert y \rvert + \lvert x_3 \rvert + \lvert x_4 \rvert + \lvert x_5 \rvert + \lvert x_6 \rvert + \lvert y + x_3 + x_4 + x_5 + x_6 \rvert)/2 | =& \left(\lvert y \rvert + \lvert x_3 \rvert + \lvert x_4 \rvert + \lvert x_5 \rvert + \lvert x_6 \rvert + \lvert y + x_3 + x_4 + x_5 + x_6 \rvert\right)/2 | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
where y = signum(x2)ceil | where y = signum(x2){{ceil|{{abs|x2/2}}}}; here "signum" is +1 or −1 depending on the sign of x2 and {{ceil|''x''}} is the ceiling function. Hahn distance for the 9 or 11 limit can also be found from this formula. | ||
It should be noted that this formula defines a {{w|Metric space|metric space distance function}} but not a norm, and hence does not define a normed vector space, making the 9-, 11- or 13-limit pitch classes into a lattice. We can modify it to | It should be noted that this formula defines a {{w|Metric space|metric space distance function}} but not a norm, and hence does not define a normed vector space, making the 9-, 11- or 13-limit pitch classes into a lattice. We can modify it to | ||
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This makes the 9.5.7.11.13 sublattice symmetrical, corresponded to even distance values from the origin, with the full lattice corresponding to all positive integer distances. | This makes the 9.5.7.11.13 sublattice symmetrical, corresponded to even distance values from the origin, with the full lattice corresponding to all positive integer distances. | ||
== Examples == | |||
{| class="wikitable" | |||
|+ style="font-size: 105%;" | Hahn distance of 5-limit intervals | |||
|- | |||
! Ratio | |||
! 5-odd-limit | |||
! 9-odd-limit | |||
! 15-odd-limit | |||
! 25-odd-limit | |||
! 27-odd-limit | |||
|- | |||
| [[6/5]] | |||
| 1 | |||
| 1 | |||
| 1 | |||
| 1 | |||
| 1 | |||
|- | |||
| [[10/9]] | |||
| 2 | |||
| 1 | |||
| 1 | |||
| 1 | |||
| 1 | |||
|- | |||
| [[16/15]] | |||
| 2 | |||
| 2 | |||
| 1 | |||
| 1 | |||
| 1 | |||
|- | |||
| [[25/24]] | |||
| 2 | |||
| 2 | |||
| 2 | |||
| 1 | |||
| 1 | |||
|- | |||
| [[27/25]] | |||
| 3 | |||
| 2 | |||
| 2 | |||
| 2 | |||
| 1 | |||
|- | |||
| [[45/32]] | |||
| 3 | |||
| 2 | |||
| 2 | |||
| 2 | |||
| 2 | |||
|- | |||
| [[75/64]] | |||
| 3 | |||
| 3 | |||
| 2 | |||
| 2 | |||
| 2 | |||
|- | |||
| [[81/80]] | |||
| 4 | |||
| 2 | |||
| 2 | |||
| 2 | |||
| 2 | |||
|- | |||
| [[135/128]] | |||
| 4 | |||
| 3 | |||
| 2 | |||
| 2 | |||
| 2 | |||
|} | |||
[[Category:Math]] | [[Category:Math]] | ||